What should be the presentation of $\mathbb Z$?

Question

In the Dummit-Foote text the definition of relation and presentation (Group theory) are introduced as:

In connection with the above definition I wounder what should be the presentation of $\mathbb Z$? Clearly enough here the generating set $S=\{1\}.$ But I'm clueless about the set of equation(s) $R_i$ in $\{1,0\}$ so that $\mathbb Z$ can be presented as:

$$\mathbb Z=\langle 1:R_i\rangle$$

Answer 1

$\mathbb Z$ is the free group on one generator - free groups have generators but no relations. You could say that the set of relations is empty.

Answer 2

$\mathbb Z$ is a free group with one generator. So $\mathbb Z$ has a single generator without any relation. In the formulation of Dummit-Foote, $m = 0$ and $$\mathbb Z = \langle \{a\} \mid \ \rangle.$$

Answer 3

It's just $\mathbb{Z}=\langle z \mid \emptyset\rangle$